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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the Steiner ellipse of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
, also called the Steiner circumellipse to distinguish it from the
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
, is the unique circumellipse ( ellipse that touches the triangle at its vertices) whose center is the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
.Weisstein, Eric W. "Steiner Circumellipse." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/SteinerCircumellipse.html Named after
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
, it is an example of a
circumconic In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld- ...
. By comparison the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. The area of the Steiner ellipse equals the area of the triangle times \frac, and hence is 4 times the area of the Steiner inellipse. The Steiner ellipse has the least area of any ellipse circumscribed about the triangle. The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid). Hence both ellipses are similar (have the same
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
).


Properties

* A Steiner ellipse is the only ellipse, whose center is the centroid S of a triangle ABC and contains the points A,B,C. The area of the Steiner ellipse is \tfrac-fold of the triangle's area. ;Proof: A) For an equilateral triangle the Steiner ellipse is the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, which is the only ellipse, that fulfills the preconditions. The desired ellipse has to contain the triangle reflected at the center of the ellipse. This is true for the circumcircle. A
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
is uniquely determined by 5 points. Hence the circumcircle is the only Steiner ellipse. B) Because an arbitrary triangle is the affine image of an equilateral triangle, an ellipse is the affine image of the unit circle and the centroid of a triangle is mapped onto the centroid of the image triangle, the property (a unique circumellipse with the centroid as center) is true for any triangle. The area of the circumcircle of an equilateral triangle is \tfrac-fold of the area of the triangle. An affine map preserves the ratio of areas. Hence the statement on the ratio is true for any triangle and its Steiner ellipse.


Determination of conjugate points

An ellipse can be drawn (by computer or by hand), if besides the center at least two conjugate points on conjugate diameters are known. In this case * ''either'' one determines by Rytz's construction the vertices of the ellipse and draws the ellipse with a suitable ellipse compass *''or'' uses an parametric representation for drawing the ellipse. Let be ABC a triangle and its centroid S. The shear mapping with axis d through S and parallel to AB transforms the triangle onto the isosceles triangle A'B'C' (see diagram). Point C' is a vertex of the Steiner ellipse of triangle A'B'C'. A second vertex D of this ellipse lies on d, because d is perpendicular to SC' (symmetry reasons). This vertex can be determined from the data (ellipse with center S through C' and B', , A'B', =c) by ''calculation''. It turns out that :, SD, =\frac\ . Or by ''drawing'': Using de la Hire's method (see center diagram) vertex D of the Steiner ellipse of the isosceles triangle A'B'C' is determined.
The inverse shear mapping maps C' back to C and point D is fixed, because it is a point on the shear axis. Hence semi diameter SD is conjugate to SC.
With help of this pair of conjugate semi diameters the ellipse can be drawn, by hand or by computer.


Parametric representation and equation

Given: Triangle \ A=(a_1,a_2),\; B=(b_1,b_2), \; C=(c_1,c_2)
Wanted: Parametric representation and equation of its Steiner ellipse The centroid of the triangle is \ S=(\tfrac,\tfrac)\ . Parametric representation: From the investigation of the previous section one gets the following parametric representation of the Steiner ellipse: *\ \vec x =\vec p(t)=\overrightarrow\; +\; \overrightarrow\; \cos t \;+\; \frac\overrightarrow\; \sin t \; , \quad 0\le t <2\pi \; . * The four vertices of the ellipse are \quad \vec p(t_0),\; \vec p(t_0\pm\frac),\; \vec p(t_0+\pi),\ where t_0 comes from :: \cot (2t_0)= \frac\quad with \quad \vec f_1=\vec,\quad \vec f_2=\frac\vec \quad (see ellipse). The roles of the points for determining the parametric representation can be changed. ''Example'' (see diagram): A=(-5,-5),B=(0,25), C=(20,0). Equation: If the origin is the centroid of the triangle (center of the Steiner ellipse) the equation corresponding to the parametric representation \vec x=\vec f_1 \cos t + \vec f_2 \sin t is *\ (xf_ - yf_)^2 + (yf_ - xf_)^2 - (f_f_ - f_f_)^2 =0 \ , with \ \vec f_i=(f_,f_)^T \ .CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt)
(PDF; 3,4 MB), p. 65.
''Example:'' The centroid of triangle \quad A=(-\tfrac\sqrt,-\tfrac),\ B=(\tfrac,-\tfrac),\ C=(\sqrt,3)\quad is the origin. From the vectors \vec f_1=(\sqrt,3)^T,\ \vec f_2=(2,0)^T \ one gets the equation of the Steiner ellipse: :9x^2+7y^2-6\sqrt xy-36=0 \ .


Determination of the semi-axes and linear eccentricity

If the vertices are already known (see above), the semi axes can be determined. If one is interested in the axes and eccentricity only, the following method is more appropriate: Let be a,b,\; a>b the semi axes of the Steiner ellipse. From Apollonios theorem on properties of conjugate semi diameters of ellipses one gets: : a^2+b^2=\vec^2+\vec^2 \ , \quad a\cdot b= \left, \det(\vec,\vec)\ \ . Denoting the right hand sides of the equations by M and N respectively and transforming the non linear system (respecting a>b>0) leads to: :a^2+b^2=M ,\ ab=N \quad \rightarrow \quad a^2+2ab+b^2=M+2N ,\ a^2-2ab+b^2=M-2N : \rightarrow\quad (a+b)^2=M+2N ,\ (a-b)^2=M-2N \quad \rightarrow\quad a+b=\sqrt ,\ a-b=\sqrt\ . Solving for a and b one gets the semi axes: * \ a=\frac(\sqrt+\sqrt) \ , \qquad b=\frac(\sqrt-\sqrt)\ , with \qquad M= \vec^2+\frac\vec^2\ , \quad N= \frac , \det(\vec,\vec), \qquad . The linear eccentricity of the Steiner ellipse is * c=\sqrt=\cdots=\sqrt\ . and the area *:F=\pi ab=\pi N=\frac \left, \det(\vec,\vec)\ One should not confuse a,b in this section with other meanings in this article !


Trilinear equation

The equation of the Steiner circumellipse in
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
is :bcyz+cazx+abxy=0 for side lengths ''a, b, c''.


Alternative calculation of the semi axes and linear eccentricity

The semi-major and semi-minor axes (of a triangle with sides of length a, b, c) have lengths :\frac\sqrt, and focal length :\frac\sqrt where :Z=\sqrt. The foci are called the '
Bickart points
'' of the triangle.


See also

* Triangle conic


References

* Georg Glaeser, Hellmuth Stachel, Boris Odehnal: ''The Universe of Conics'', Springer 2016, {{ISBN, 978-3-662-45449-7, p.383 Curves defined for a triangle Conic sections